No, you wouldn't get the message until approaching Vmo.
Thanks for the input. I can remember seeing DRAG REQUIRED at times where I was nowhere near VMO-16. Probably most of us have seen the DRAG REQUIRED message when we were nowhere near 349 knots.
The FCOM I have only mentions that the DRAG REQUIRED message will appear at 349 knots (VMO/MMO-16) on a VNAV descent while attempting to stay in VNAV PTH. Nowhere else in the FCOM is it discussed.
https://drive.google.com/file/d/0B-W...ew?usp=sharing
There obviously must be more than this one situation (349 knots) that would set off the DRAG REQUIRED message, but I can't find it in any material in my possession. It seems that there are a number of differeing opinions as to when you would see this message. I'm just trying to see if there's anything other than the above section that would shed some light on this.